Crystallographic Haar wavelets
Let \(\Gamma\) be a \(d\)-dimensional crystallographic group and let \(a:\,{\mathbb R}^d \to {\mathbb R}^d\) be an expanding affine map. By definition, \((\Gamma,a)\)-crystallographic multiwavelets form a finite set of functions \(\{\psi^1,\ldots, \psi^L\}\), which generate an orthonormal basis, a Riesz basis or a Parseval frame for \(L^1({\mathbb R}^d)\) under the action of \(\Gamma\) and powers of \(a\), taking the place of translations and dilations in classical wavelets. This paper on crystallographic Haar-type multiwavelets is based on self-replicant tilings (see \textit{M. del C. Moure} [Discrete Comput. Geom. 42, No. 4, 722--739 (2009; Zbl 1186.28007)]), reptiles associated to crystallographic groups (see \textit{G. Gelbrich} [Geom. Dedicata 51, No. 3, 235--256 (1994; Zbl 0807.05022)]) and characterizations of scaling functions for Haar-type multiresolution analyses (see [\textit{K. Gröchenig} and \textit{W.R. Madych}, IEEE Trans. Inf. Theory 38, 556--568 (1994)]). The authors characterize scaling function vectors for Haar-type crystallographic \((\Gamma,a)\)-multiresolution analyses. Then corresponding \((\Gamma,a)\)-crystallographic multiwavelets of Haar type are constructed. Examples are presented in the cases \(d=2\) and \(d=3\).
crystallographic multiresolution analysis
scientific article; zbMATH DE number 5996269
scientific article